# forward contract on a defaultable zero-coupon bond

I'am trying to calculate the price of a forward on a defaultable zero-coupon bond. It is also true that the price will be given by Price a forward contract on a zero-coupon bond ? I guess the defautability of the bond shoud result in higher price for the forward than in the standard case.

Thanks.

Based on notations in this question, assuming the market value recovery mechanism, the pre-default value at time $$T_1$$ of a zero-coupon bond with maturity $$T_2$$, where $$T_1 < T_2$$, is given by \begin{align*} P(T_1, T_2) = E\Big(e^{-\int_{T_1}^{T_2}(r_s +(1-R)h_s)ds}\,\big|\, \mathscr{F}_{T_1}\Big). \end{align*} Let $$B_t=e^{\int_0^t r_s ds}$$ be the credit risk free money-market account value at time $$t$$. The pre-default forward price $$K$$ determined at time $$t$$, for $$0\le t \le T_1$$, is a value such that \begin{align*} 0 &= E\Big(\pmb{1}_{\tau>T_1}\frac{B_t}{B_{T_1}}(K-P(T_1, T_2)) \,|\,\mathscr{G}_t\Big)\\ &=\pmb{1}_{\tau>t}E\left(\Big(K e^{-\int_t^{T_1}(r_s+h_s) ds} - e^{-\int_t^{T_1}(r_s+h_s) ds-\int_{T_1}^{T_2}(r_s +(1-R)h_s)ds} \Big) \,|\,\mathscr{F}_t\right)\\ &=\pmb{1}_{\tau>t}E\left(\Big(K e^{-\int_t^{T_1}(r_s+h_s) ds} - e^{-\int_t^{T_2}(r_s+h_s) ds+\int_{T_1}^{T_2}Rh_sds} \Big) \,|\,\mathscr{F}_t\right). \end{align*} That is, \begin{align*} K = \frac{E\Big(e^{-\int_t^{T_2}(r_s+h_s) ds+\int_{T_1}^{T_2}Rh_sds} \,|\,\mathscr{F}_t\Big)}{E\Big(e^{-\int_t^{T_1}(r_s+h_s) ds} \,|\,\mathscr{F}_t\Big)}. \end{align*}
Your observation appears correct if you assume that the interest rate is defined by $$r_t+h_t$$ in the standard case.
• Shouldn't the second expectation be conditional on $\mathscr{G}_t$? Thanks.